Consumption-Based Asset Pricing with Higher Cumulants

Transcription

1 Review of Economic Studies (2013) 80, doi: /restud/rds029 The Author Published by Oxford University Press on behalf of The Review of Economic Studies Limited. Advance access publication 10 September 2012 Consumption-Based Asset Pricing with Higher Cumulants IAN W. R. MARTIN Graduate School of Business, Stanford University First version received January 2009; final version accepted April 2012 (Eds.) I extend the Epstein Zin-lognormal consumption-based asset-pricing model to allow for general i.i.d. consumption growth. Information about the higher moments equivalently, cumulants of consumption growth is encoded in the cumulant-generating function. I use the framework to analyse economies with rare disasters, and argue that the importance of such disasters is a double-edged sword: parameters that govern the frequency and sizes of rare disasters are critically important for asset pricing, but extremely hard to calibrate. I show how to sidestep this issue by using observable asset prices to make inferences without having to estimate higher moments of the underlying consumption process. Extensions of the model allow consumption to diverge from dividends, and for non-i.i.d. consumption growth. Keywords: Consumption-based asset pricing, Rare disasters, Cumulants JEL Codes: G10, E44. The combination of power utility and i.i.d. lognormal consumption growth makes for a tractable benchmark model in which asset prices and expected returns can be found in closed form. This article demonstrates that the lognormality assumption can be dropped without sacrificing tractability, thereby allowing for straightforward and flexible analysis of the possibility that, say, consumption is subject to rare disasters. There has recently been considerable interest in reviving the idea of Rietz (1988) that the presence of such disasters, or fat tails more generally, can help to explain asset pricing phenomena such as the riskless rate, equity premium and other puzzles (Barro, 2006; Jurek, 2008). Here, I take a different line, closer in spirit to Weitzman (2007), and argue that the importance of rare, extreme events is a double-edged sword: those model parameters that are most important for asset prices, such as disaster parameters, are also the hardest to calibrate, precisely because the disasters in question are rare. Working under the assumptions that there is a representative agent with Epstein Zin preferences (Epstein and Zin, 1989) and that consumption growth is i.i.d., Section 1 shows that the equity premium, riskless rate, consumption wealth ratio and mean consumption growth (the fundamental quantities ) can be simply expressed in terms of the cumulant-generating function (CGF). CGFs crop up elsewhere in the literature; one contribution of this article is to demonstrate how neatly they dovetail with the standard consumption-based asset-pricing approach. Importantly, the framework allows for the possibility of disasters, but is agnostic about whether or not they occur. The expressions derived relate the fundamental quantities directly to the cumulants (equivalently, moments) of consumption growth. I show, for example, how the precautionary savings effect, which influences the riskless rate in a lognormal model, 745

2 746 REVIEW OF ECONOMIC STUDIES can be generalized in the presence of higher cumulants. By shifting the focus from moments to cumulants, I retain tractability without needing to truncate Taylor expansions (as in, say, Kraus and Litzenberger, 1976), to avoid the critique of Brockett and Kahane (1992). Section 2 illustrates the framework by investigating a continuous-time model featuring rare disasters, and shows that the model s predictions are sensitively dependent on the calibration assumed. As a stark example, take a consumption-based model in which the representative agent has relative risk aversion equal to 4. Now add to the model a certain type of disaster that strikes, on average, once every 1,000 years, and reduces consumption by 64 per cent. (Barro (2006) documents that Germany and Greece each suffered such a fall in per capita real GDP during the Second World War.) The introduction of this disaster drives the riskless rate down by 5.9 percentage points and increases the equity premium by 3.7%. 1 Very rare, very severe events exert an extraordinary influence on the benchmark model, and we do not expect to estimate their frequency and intensity directly from the data. I document this more formally in Section 2.2, where I present the results of a GMM exercise as in Hansen and Singleton (1982). I use samples consisting of 100 years of simulated data in the model economy with disasters, and show that in such a relatively short sample, GMM leads to biased and extremely inaccurate estimators of the true population parameters. In economies with still fatter tails, GMM may not be valid even asymptotically. The remainder of this article is devoted to finding ways around this disheartening fact. We can, for example, detect the influence of disaster events indirectly, by observing asset prices. I argue, therefore, that the standard approach calibrating a particular model and trying to fit the fundamental quantities is not the way to go. I turn things round, viewing the fundamental quantities as observables, and making inferences from them. It then becomes possible to make non-parametric statements that are robust to the details of the consumption growth process. In this spirit, I derive, in Section 3, sharp and robust restrictions on preference parameters that are valid in any Epstein Zin-i.i.d. model that is consistent with the observed fundamentals. The key idea is to exploit an important property of CGFs: they are always convex. The results restrict the time-preference rate, ρ, and elasticity of intertemporal substitution, ψ, to lie in a certain subset of the positive quadrant. (See Figure 7.) These parameters are of central importance for financial and macroeconomic models. The restrictions depend only on the Epstein Zin-i.i.d. assumptions and on observed values of the fundamental quantities, and not, for example, on any assumptions about the existence, frequency or size of disasters. They are complementary to econometric or experimental estimates of ψ and ρ, and are of particular interest because there is little agreement about the value of ψ. (Campbell (2003) summarizes the conflicting evidence.) I also show how good-deal bounds (Cochrane and Saá-Requejo, 2000) can be used to provide upper bounds on risk aversion, based once again on the fundamental quantities, without calibrating a consumption process. Why work with cumulants and CGFs, rather than with moments and moment-generating functions? Aside from the fact that they are what drop out of the algebra, cumulants are in many respects more intuitive than moments. The first cumulant is the mean; the second is variance, σ 2 ; the third and fourth cumulants are skewness times σ 3 and excess kurtosis times σ 4, respectively. These are easier to grasp intuitively than the corresponding non-central moments. A second, equally important, reason is that CGFs have the convexity property mentioned above. Momentgenerating functions (as exponentials of CGFs) are convex too, but this is a much cruder, and therefore less useful, fact. 1. The effect is smaller with Epstein Zin preferences if the elasticity of substitution is greater than 1, but even with an elasticity of intertemporal substitution equal to 2, the riskless rate drops by 3.5%.

3 IAN MARTIN CONSUMPTION-BASED ASSET PRICING 747 Section 4 extends the analysis in two directions. The majority of the article sets dividends equal to consumption or adopts the highly tractable approach, taken by Campbell (1986, 2003) and also advocated by Abel (1999), of modelling dividends as a power of consumption. However, several authors have argued for the importance of allowing log consumption and log dividends to be imperfectly correlated. 2 Section 4.1 shows how the CGF approach can be extended to allow for this possibility, and presents a heterogeneous-agent model as a motivating application. Consistent with the argument of the rest of the article, I find that heterogeneity is a double-edged sword. The good news is that heterogeneity interacts well with disasters in the sense that it can potentially give a huge boost to risk premia. The bad news is that heterogeneity only matters to the extent to which it occurs at times of aggregate disaster, so the fundamental empirical difficulty highlighted in the rest of the paper is not avoided. Again, though, it is possible to make statements that are robust to what is going on in the tails. Finally, although presumably the framework is a good approximation to reality over time horizons long enough that the economy looks roughly i.i.d., it is of interest to weaken the i.i.d. assumption, and I do so in Section 4.2. When not included in the body of the article, proofs are in the appendix. Related literature. Campbell and Cochrane (1999) and Bansal and Yaron (2004) modify the textbook model along different dimensions, but take care to remain in a conditionally lognormal environment. This article explores different features, and implications, of the data, so is complementary to their work. It would, of course, be interesting to extend these papers by allowing for the possibility of jumps, but doing so would obscure the main point of this article. Various authors have presented analytical solutions in similar models. Eraker (2008) prices assets from the perspective of an Epstein Zin representative agent, but relies on a loglinearization of the return on aggregate wealth for tractability. This approximation is likely to be particularly problematic in a disaster model in which aggregate wealth may experience severe declines. Bonomo et al. (2011) provide analytical pricing formulas in a long-run risks environment with generalized disappointment aversion, although their focus is not on rare disasters. Martin (2011, 2013) uses CGFs in multi-asset models in which consumption growth is not i.i.d. A large body of literature applies Lévy processes to derivative pricing (Carr and Madan, 1998, Cont and Tankov, 2004) and portfolio choice (Cvitanić, Polimenis, and Zapatero, 2005, Aït-Sahalia, Cacho-Diaz, and Hurd, 2006). Lustig, Van Nieuwerburgh, and Verdelhan (2008) present estimates of the wealth consumption ratio. Backus, Foresi, and Telmer (2001) derive expressions relating cumulants to risk premia, though their approach is very different from that taken here. Garcia, Luger, and Renault (2003) expand the range of assets, using options prices to obtain information about preference parameters, though they work in a conditionally lognormal framework. Backus, Chernov, and Martin (2011) explore the evidence for disasters in option prices, but leave unresolved the finding of Coval and Shumway (2001) that all option prices, at-the-money options included, appear mysteriously high. Why might this be? Perhaps because at-the-money put options are assets with potentially enormous payoffs in the very bad times we do not have enough of in the data. (If so, at-the-money calls would also look expensive, by putcall parity.) Julliard and Ghosh (2008) argue that the cross-section of asset price data is hard to square with disaster explanations of the equity premium. Consistent with the above discussion, 2. For example, Cecchetti, Lam, and Mark (1993), Bonomo and Garcia (1996), Campbell and Cochrane (1999), Longstaff and Piazzesi (2004), and Bansal and Yaron (2004). With the exception of Longstaff and Piazzesi (2004), consumption and dividends are not cointegrated in any of these papers.

4 748 REVIEW OF ECONOMIC STUDIES their parameter estimates have large standard errors. They also carry out a Generalized Empirical Likelihood estimation whose results are similar to those of Section ASSET PRICING AND THE CGF Define G t logc t /C 0 and write G G 1. I make two assumptions. A1 There is a representative agent with Epstein Zin preferences, time preference rate ρ, relative risk aversion γ, and elasticity of intertemporal substitution ψ. A2 The consumption growth, logc t /C t 1, of the representative agent is (or is perceived to be) i.i.d., and the CGF of G (defined below) exists on a neighbourhood of [ γ,1]. 3 Assumption A1 allows risk aversion γ to be disentangled from the elasticity of intertemporal substitution ψ. To keep things simple, those calculations that appear in the main text restrict to the power utility case in which ψ is constrained to equal 1/γ ; in this case, the representative agent maximizes ρt C1 γ t E e 1 γ t=0 if γ =1, or E e ρt logc t if γ =1. t=0 Cogley (1990) and Barro (2009) present evidence in support of A2 in the form of variance-ratio statistics close to one, on average, across 9 (Cogley) or 19 (Barro) countries. We need expected utility to be well defined in that C1 γ t E e ρt < if γ =1. (1) 1 γ t=0 I discuss this requirement further below. Consider an asset that pays dividend stream {D t } t 0. The Euler equation relates the price of an asset this period, P 0, to the payoff next period, P 1 +D 1. Expectations are calculated with respect to the measure perceived by the representative agent: P 0 =E 0 ( ( ) γ e ρ C1 (D 1 +P 1 )). C 0 Iterating forward and imposing a no-bubble condition, we have the familiar equation P 0 =E ( t=1 ( ) ) γ e ρt Ct D t. C 0 Suppose that D t (C t ) λ for some constant λ. Ifλ=0 then the asset is a riskless bond; if λ=1 then it is the wealth portfolio that pays consumption as its dividend. As suggested by 3. If not, the consumption-based asset-pricing approach is invalid. This assumption implies that all cumulants, and hence all moments, of G are finite. See Billingsley(1995, Section 21).

5 IAN MARTIN CONSUMPTION-BASED ASSET PRICING 749 Campbell (1986, 2003) and Abel (1999), it is possible to view values λ>1 as a tractable way of modelling levered claims. Writing P 0 for the price of this asset at time 0, we have ( ) γ P 0 =E e ρt Ct C λ t =D 0 e ρt Ee (λ γ )G ( t =D 0 e ρt Ee (λ γ )G) t. (2) C t=1 0 t=1 t=1 The last equality follows from the assumption that log consumption growth is i.i.d. To make further progress, I now introduce a pair of definitions. Definition 1. Given some arbitrary random variable, G, the moment-generating function m(θ) and cumulant-generating function or CGF c(θ) are defined by for all θ for which the expectations are finite. m(θ) Eexp(θG) c(θ) logeexp(θg), Here, G is an annual increment of log consumption, G=logC t+1 logc t. Notice that c(0)=0 for any growth process and that c(1) is equal to log mean gross consumption growth, so we will want c(1) 2%. The CGF summarizes information about the cumulants (or, equivalently, moments) of G. We can expand c(θ) as a power series in θ, κ n θ n c(θ)=, (3) n! n=1 which defines κ n as the n-th cumulant of log consumption growth. Some algebra shows that the first few cumulants are familiar: κ 1 μ is the mean, κ 2 σ 2 the variance, κ 3 /σ 3 the skewness, and κ 4 /σ 4 the excess kurtosis of log consumption growth. Knowledge of the cumulants of a random variable implies knowledge of the moments, and vice versa. With this definition, (2) becomes P 0 =D 0 e [ρ c(λ γ )]t e [ρ c(λ γ )] =D 0. 1 e [ρ c(λ γ )] t=1 It is convenient to define the log dividend yield d/p log(1+d 0 /P 0 ). Then, d/p=ρ c(λ γ ). Two special cases are of particular interest. The first is λ=0, in which case the asset in question is the riskless bond, whose dividend yield is the riskless rate R f. Again, it is convenient to work with the log riskless rate, r f =log(1+r f ). The above calculation shows that r f =ρ c( γ ). The second is λ=1, in which case the asset pays consumption as its dividend, and can therefore be interpreted as aggregate wealth. The dividend yield is then the consumption wealth ratio; when λ=1, I write c/w in place of d/p. This calculation also shows that the necessary restriction on consumption growth for the expected utility to be well defined in (1) is that ρ c(1 γ )>0, or equivalently that the consumption wealth ratio is positive. The gross return on the λ-asset is 1+R t+1 = D t+1 +P t+1 = P ( t+1 1+ D ) t+1 = D ( t+1 e ρ c(λ γ )), P t P t P t+1 D t so the expected gross return is ( (Ct+1 ) ) λ 1+ER t+1 =E e ρ c(λ γ ) =e ρ c(λ γ )+c(λ). C t

6 750 REVIEW OF ECONOMIC STUDIES Once again, it is more convenient to work with log expected gross return, er log(1+ ER t+1 )=ρ +c(λ) c(λ γ ). Finally, I define the risk premium rp=er r f. The following result summarizes and extends the above calculations by expressing the riskless rate, dividend yield, and risk premium in terms of the CGF in the Epstein Zin case. Result 1. Defining ϑ (1 γ )/(1 1/ψ), we have r f = ρ c( γ )+c(1 γ )(1 1/ϑ) (4) d/p = ρ c(λ γ )+c(1 γ )(1 1/ϑ) (5) rp = c(λ)+c( γ ) c(λ γ ). (6) The Gordon growth model holds (note that c(λ)=loge(d t+1 /D t )): d/p=rp+r f c(λ). (7) The consumption wealth ratio c/w is given by (5) with λ=1. As in the power utility case, these expressions are well defined so long as c/w>0. Equation (6) shows that the elasticity of intertemporal substitution does not affect the risk premium. It also shows that the CGF of the driving consumption process must have a significant amount of convexity over the range [ γ,λ] to generate an empirically reasonable risk premium. These expressions can be written out as power series using (3). In the power utility case, for example, Equation (4) implies that r f =ρ +κ 1 γ κ 2 2 γ 2 + κ 3 3! γ 3 κ 4 4! γ 4 +higher order terms. By definition of the first four cumulants, this can be rewritten as r f =ρ +μγ 1 2 σ 2 γ 2 + skewness σ 3 γ 3 3! excess kurtosis σ 4 γ 4 +higher order terms. (8) 4! If consumption growth is lognormal, skewness, excess kurtosis, and all higher cumulants are zero, so this reduces to the familiar r f =ρ +μγ σ 2 γ 2 /2. More generally, the riskless rate is low if mean log consumption growth μ is low (an intertemporal substitution effect); if the variance of log consumption growth σ 2 is high (a precautionary savings effect); if there is negative skewness; or if there is a high degree of kurtosis. Similarly, the dividend yield is d/p = ρ +μ(γ λ) 1 2 σ 2 (γ λ) 2 + skewness σ 3 (γ λ) 3 3! excess kurtosis σ 4 (γ λ) 4 +higher order terms, 4! and the risk premium (in either the power utility or the Epstein Zin case) is rp = λγ σ 2 + skewness σ 3( λ 3 γ 3 (λ γ ) 3) + 3! excess kurtosis + σ 4( λ 4 +γ 4 (λ γ ) 4) +higher order terms. 4!

7 IAN MARTIN CONSUMPTION-BASED ASSET PRICING 751 To understand what happens with Epstein Zin preferences, it is helpful to focus on the case λ=1,γ >1. (The logic is the same if λ =1; some signs are reversed if γ<1.) The coefficients on κ n /n! in the power series expansions are r f : ( 1) n+1 γ n +( 1) n (γ 1) n (1 1/ϑ) c/w: ( 1) n+1 (γ 1) n /ϑ rp: 1+( 1) n γ n +( 1) n+1 (γ 1) n c(1): 1 The Gordon growth formula (7) implies that the n-th coefficient for c/w is equal to the n-th coefficient for r f, plus that for rp, minus that for (log) expected consumption growth, c(1). So it suffices to understand the comparative statics of the risk premium and of the riskless rate. The comparative statics of the risk premium are the same in the power utility and Epstein Zin cases. The n-th coefficient is 1+( 1) n γ n +( 1) n+1 (γ 1) n. The third of these terms is smaller in magnitude than the second, but has the opposite sign, so exerts an offsetting effect. Thus the n-th coefficient is positive for even n 2: the risk premium is increasing in variance and higher even cumulants. For odd n 3, the coefficient is negative, so the risk premium is decreasing in skewness and higher odd cumulants. The comparative statics of the riskless rate depend on both ψ and γ. With power utility, the n-th coefficient in the expansion of r f is ( 1) n+1 γ n. This is positive if n is odd and negative if n is even, leading to the comparative statics discussed below Equation (8). There is no offsetting term in (γ 1) n, so the riskless rate is more sensitively dependent on higher cumulants than the risk premium. In the Epstein Zin case, we gain an extra term ( 1) n (γ 1) n (1 1/ϑ). If ψ =1 then 1/ϑ =0 and the n-th coefficient is ( 1) n+1 γ n +( 1) n (γ 1) n, which does have the offsetting term, resulting in a riskless rate that is less sensitively dependent on the higher cumulants than in the power utility case. More generally, if 1/ϑ <0 if γ>1 and ψ>1 then for small n the term ( 1) n (γ 1) n (1 1/ϑ) may even dominate. For large n, though, the first term always prevails, so the riskless rate depends less sensitively on high cumulants than it does in the power utility case. We are now in a position to understand the comparative statics of the consumption wealth ratio. With power utility, the riskless rate is the dominant influence: it is increasing in odd cumulants and decreasing in even cumulants, and the consumption wealth ratio inherits that property. With Epstein Zin preferences and 1 1/ϑ > 0 i.e. ψ > 1/γ, assuming γ > 1 the effects of higher cumulants on the riskless rate are muted. If ψ =1, the movements of the riskless rate exactly offset the movements of the risk premium, so that the consumption wealth ratio is constant. If ψ>1, so that 1/ϑ <0, then the risk premium effect is dominant; the consumption wealth ratio is then increasing in even cumulants and decreasing in odd cumulants. Put differently, larger even cumulants lead to a lower wealth consumption ratio; this is an important component of Bansal and Yaron s (2004) long-run risk model. Equations (4) (6), together with the Gordon growth model (7), provide another way to look at a point made by Kocherlakota (1990). In principle, given sufficient asset price and consumption data, we could determine the riskless rate, the risk premium, and CGF c( ) to arbitrary accuracy. Since γ is the only preference parameter that determines the risk premium, it could be calculated from (6), given knowledge of c( ). On the other hand, knowledge of the riskless rate leaves ρ and ψ indeterminate in Equation (4), even given knowledge of γ and c( ). So the time discount rate and elasticity of intertemporal substitution cannot be disentangled on the basis of the four fundamental quantities.

8 752 REVIEW OF ECONOMIC STUDIES 2. THE CONTINUOUS-TIME CASE In continuous time, the analogue of the i.i.d. growth assumption is that the log consumption path, G t, of the representative agent follows a Lévy process. If so, Ee θg t = ( Ee θg) t for arbitrary t 0. Given this property, we have the following result in the limit as the period length dt (which was equal to one in the discrete-time calculations) goes to zero. Result 2 (The continuous-time case). The instantaneous riskless rate, R f, dividend yield, D/P, and instantaneous risk premium on aggregate wealth, RP, are R f = ρ c( γ )+c(1 γ )(1 1/ϑ) D/P = ρ c(λ γ )+c(1 γ )(1 1/ϑ) RP = c(λ)+c( γ ) c(λ γ ). The Gordon growth model holds: D/P =R f +RP c(λ) A concrete example: disasters In this section, I show how to derive a convenient continuous-time version of Barro (2006), and show that the predictions of an i.i.d. disaster model are sensitively dependent on the parameter values assumed. Suppose that log consumption follows a jump-diffusion N(t) G t = μt +σ B B t + Y i i=1 where B t is a Brownian motion, N(t) is a Poisson counting process with parameter ω, and Y i are i.i.d. random variables. The CGF is c(θ)=logm(θ), where m(θ)=ee θg 1 =e μθ Ee σ BθB1 Ee θ N(1) i=1 Y i. Separating the expectation into two separate products is legitimate since the Poisson jumps and Y i are independent of the Brownian component B t. The middle term is the expectation of a lognormal random variable: Ee θσ BB 1 =e σ B 2θ 2 /2. The final term is slightly more complicated, but can be evaluated by conditioning on the number of jumps that take place before t =1: N(1) Eexp θ { } Y i = e ω ω n n Eexp θ Y i n! i=1 0 1 e ω ω n = [Eexp{θY 1 }] n n! 0 = exp { ω ( m Y1 (θ) 1 )}, Finally, c(θ)= μθ +σb 2θ 2 /2+ω ( m Y1 (θ) 1 ), so the cumulants κ n (G)=c (n) (0) are μ+ω EY n=1 κ n (G)= σ B 2 +ω EY 2 n=2 ω EY n n 3.

9 IAN MARTIN CONSUMPTION-BASED ASSET PRICING 753 (a) (b) Figure 1 (a) The CGF with (solid) and without (dashed) jumps. The figure assumes that ρ =0.03 and γ =4. (b) Zooming out. Without jumps, we need enormously high γ to avoid the riskless rate and equity premium puzzles Figure 2 The risk premium. The figure assumes that γ =4 Take the case in which Y N( b,s 2 ); b is assumed to be greater than zero, so the jumps represent disasters. The CGF is then c(θ)= μθ σ 2 B θ 2 +ω(e θb+ 1 2 θ 2 s 2 1). (9) Figure 1a plots the CGF (9) against θ. I choose parameters according to Barro s (2006) baseline calibration γ =4,σ B =0.02,ρ=0.03, μ=0.025,ω=0.017 and set b=0.39 and s=0.25 to match the mean and variance of the distribution of jumps used in the same paper. I also plot the CGF that results in the absence of jumps (ω =0). In the latter case, I adjust the drift of consumption growth to keep mean log consumption growth constant; in the figure, this means that the two curves are tangent at the origin. Zooming out on Figure 1a, we obtain Figure 1b, which further illustrates the equity premium and riskless rate puzzles. With jumps, the CGF is visible at the right-hand side of the figure; the CGF explodes so quickly as θ declines that it is only visible for θ greater than about 5. The jump-free lognormal CGF has incredibly low curvature. For a realistic riskless rate and equity premium, the model requires a risk aversion above 80.

10 754 REVIEW OF ECONOMIC STUDIES TABLE 1 The impact of different assumptions about the distribution of disasters ω b s R f C/W RP R f C/W RP Baseline case High ω Low ω High b Low b High s Low s μ=0.025, σ =0.02. Unasterisked group assumes power utility, ρ =0.03, γ =4. Asterisked group assumes Epstein Zin preferences, ρ =0.03, γ =4, ψ =1.5. TABLE 2 The impact of approximating the disaster model by truncating at the n-th cumulant n R f C/W RP R f C/W RP 1 deterministic lognormal true model Unasterisked group assumes power utility; asterisked group assumes Epstein Zin preferences. All parameters as in baseline case of Table 1. The riskless rate, consumption wealth ratio, and mean consumption growth can be read directly off the graph, as indicated by the arrows in Figure 1a. The risk premium can be calculated from these via the Gordon growth formula, or read directly off the graph, as in Figure 2, by drawing a line from ( γ,c( γ )) to (1,c(1)) and another from (1 γ,c(1 γ )) to (0,0). The midpoint of the first line lies above the midpoint of the second by convexity of the CGF. The risk premium is twice the distance from one midpoint to the other. The standard lognormal model predicts a counterfactually high riskless rate: in Figure 1a, this is reflected in the fact that the no-jumps CGF lies well below ρ for reasonable values of θ. Similarly, the standard lognormal model predicts a counterfactually low equity premium: the no-jump CGF is practically linear over the range [ γ,1]. Conversely, the disaster CGF has a shape that allows it to match observed fundamentals closely. Table 1 shows how changes in the calibration of the distribution of disasters affect the fundamental quantities. I consider the power utility case with ρ =0.03 and γ =4, and the Epstein Zin case with the same time-preference rate and risk aversion but higher elasticity of intertemporal substitution, ψ = 1.5. The model s predictions are sensitively dependent on the parameter values. Small changes in any of ω, b,ors have large effects on the equity premium (and, with power utility, on the riskless rate; this effect is muted in the Epstein Zin case). Given that these parameters are hard to estimate disasters happen very rarely this is problematic. Table 2 investigates the consequences of truncating the CGF at the n-th cumulant. When n=2, this is equivalent to making a lognormality assumption, as noted above. With n=3, it can be thought of as an approximation that accounts for the influence of skewness; n=4 also allows for kurtosis. As is clear from the table, even calculations based on fourth- or fifth-order approximations do not fully capture the impact of disasters.

11 2.2. A GMM exercise IAN MARTIN CONSUMPTION-BASED ASSET PRICING 755 What would estimates of ρ and γ look like if the baseline disaster model were a literal description of reality? This section carries out a GMM exercise using the baseline calibration. I simulate 100 years of annual consumption data, back out asset prices and returns using the above results, and estimate the parameters ρ and γ from the sample analogues of the moment conditions 4 [ ( ) [ γ ( ) γ E e ρ Ct+1 (R t+1 R f,t+1 )] =0 and E e ρ Ct+1 ( ) ] 1+Rf,t+1 =1. (10) C t C t These equations apply in the power utility case; the conclusions of this section also apply in the Epstein Zin case if ρ is replaced by ρ ρ +(1 1/ϑ) c(1 γ ). The moment conditions (10) identify ρ and γ in the model; see the Appendix for a proof. Explicitly, the estimates ρ and γ are computed by solving T ( ) γ Ct+1 (R t+1 R f,t+1 )=0 t=1 for γ, and using the result to determine ρ from 1 T T t=1 C t ( ) γ e ρ Ct+1 ( ) 1+Rf,t+1 =1. C t I show in the Supplementary Appendix that the standard regularity conditions hold in this calibration, so that GMM estimates are consistent and asymptotically Normal. It turns out, though, that 100 years is not enough data for these asymptotic results to apply even approximately. Figure 3 shows what happens when the GMM procedure is repeated 100,000 times. Each black dot represents an estimate ( ρ, γ ) generated from 100 years of annual data simulated in the baseline disaster calibration; for clarity, the left panel excludes 7 estimates for which γ lies above 300 (with a maximum of 416). In each of the 100,000 histories, I also compute the mean realized equity premium and mean consumption growth. Across the 100,000 histories, the sample means of these variables line up perfectly with their population counterparts, at 5.7% and 2.0% respectively, and both have a standard deviation of 0.5%. The mean estimate (ρ,γ ) =( 0.086,29.5), generated by averaging the resulting 100,000 estimates ( ρ, γ ), is marked with a white dot. (For comparison, Kocherlakota (1996) carries out the same exactly identified GMM exercise in real-world data. In my notation, he estimates ρ = and γ =17.95.) The true value (ρ,γ)=(0.03,4) is also marked with a white dot. The standard deviation of the estimates ρ is 0.426; the standard deviation of the estimates γ is 43.5; and the correlation between ρ and γ is The dashed ellipses show confidence regions within which 50% (small ellipse) and 95% (large ellipse) of the sample points would lie if the data were Normally distributed. Larger estimates of risk aversion tend to be associated with smaller estimates of the time preference rate: the procedure is struggling to match the data by increasing risk aversion, at the cost of having to assuming extreme patience in order to match the riskless 4. In this section, I assume that λ=1, so we consider the unlevered consumption claim. If the exercise is repeated with levered assets, λ>1, the results discussed below apply with even more force: the small-sample properties of the various estimators are dramatically worse. This suggests that the fact that at-the-money options appear so expensive (Coval and Shumway, 2001) might be expected to remain a puzzle even longer than the equity premium puzzle.

12 756 (a) REVIEW OF ECONOMIC STUDIES (b) Figure 3 Each of the 100,000 small black dots represents a GMM estimate of (ρ,γ ) from 100 years of simulated annual data in the disaster model. The ellipses indicate confidence regions in which 50% (inner ellipse) and 95% (outer ellipse) of the mass of the distribution of ( ρ, γ ) would lie if ( ρ, γ ) were Normally distributed. Dotted lines in each panel indicate the model-free parameter restrictions derived in Section 3: just 10 of the estimates ( ρ, γ ) lie in the lower admissible region, visible in the right-hand panel, in which γ < 1 rate. Finally, dotted lines indicate model-free parameter restrictions that will be derived in the next section. The figures demonstrate three features of GMM estimation in the disaster calibration. First, there is extraordinary dispersion in the estimates. Estimates of γ extend up to more than 400. Estimates of ρ, the time discount rate, range between about 2 and 6: in time-discount-factor ρ, these correspond to β = 7.39 and β = , respectively. Second, ρ terms, writing β = e is a downward-biased estimator of ρ and γ is an upward-biased estimator of γ. Third, the estimates ( ρ, γ ) are far from Normally distributed when using 100 years of data: the confidence ellipses utterly fail to capture the shape of the distribution. (Contrast Figure 3a with Figure 4b.) A formal test of Normality is superfluous, since amongst the 100,000 sample points there are several estimates ρ that lie more than 10 standard deviations above the mean ρ (i.e. are larger than about 4.2). The probability of a 10-sigma event under the Normal distribution is about 10 23, so even one such occurrence in 100,000 samples would be sufficient to reject the hypothesis of Normality at the % level. These conclusions are not driven by extreme outliers. Figure 4a shows the result of trimming the data by discarding the largest 5% of pairs ordered by γ. The GMM estimates are still biased (ρ,γ ) = ( 0.076,23.6) and the confidence ellipses are still large. As a sanity check, and as a contrast with the disaster calibration, Figure 4b shows what happens in a lognormal world. It reports the results of carrying out the same exercise in a calibration without disasters (ω = 0), with σb adjusted upwards so that the risk premium remains constant. As one would expect, GMM provides an accurate estimate of the underlying population parameters, and the sample distribution is approximately Normal. With more extreme specifications of disaster sizes, even worse behavior is possible (Kocherlakota, 1997). If the random variables inside the expectations in (10) do not have finite second moment, the GMM estimators are not even asymptotically Normal. In the case of the moment condition for the riskless bond, we require c( 2γ ) to be finite; in the case of the consumption claim, we require c(2 2γ ) to be finite. If the former is finite, the latter is too, so in summary the GMM approach is only valid if c( 2γ ) is finite. This is not assured by the assumptions that ensure finite utility, or a finite consol price. Below, I construct an example for [14:22 4/4/2013 rds029_online.tex] RESTUD: The Review of Economic Studies Page:

13 IAN MARTIN (a) CONSUMPTION-BASED ASSET PRICING 757 (b) Figure 4 (a) The message of the previous figure is unaltered if the most extreme 5% of realizations ( ρ, γ ) is discarded. (b) GMM estimates of (ρ,γ ) from 100 years of simulated annual data in a lognormal model which the GMM approach fails not only in finite samples, but even asymptotically: Figure 6b plots the CGF of (an extreme case of) such a distribution. 3. RESTRICTIONS ON PREFERENCE PARAMETERS I now assume that the riskless rate, consumption wealth ratio, and risk premium (and hence expected consumption growth, via the Gordon growth model (7)) are observable and take the values given in Table 3.5 The observables provide us with information about the shape of the CGF; this section shows how to use this information to derive restrictions on the preference parameters that must hold in any Epstein Zin/i.i.d. model, no matter what is going on in the tails. The restrictions are of particular interest when there are small sample biases, as in the previous section, and they continue to be valid when GMM breaks down entirely, as discussed at the end of the previous section. TABLE 3 Assumed values of the observables riskless rate risk premium consumption wealth ratio rf rp c/w For example, rf = ρ c( γ ) in the power utility case, so observation of the riskless rate tells us something about ρ and something about the value taken by the CGF at γ. Similarly, observation of the consumption wealth ratio tells us something about ρ and something about the value taken by the CGF at 1 γ. Next, c(1) = loge(c1 /C0 ) is pinned down by the Gordon 5. As noted in the above GMM exercise, the equity premium can be accurately estimated, even in a world with rare disasters: in the baseline calibration, the equity premium conditional on no disasters is very close to the unconditional equity premium. [14:22 4/4/2013 rds029_online.tex] RESTUD: The Review of Economic Studies Page:

14 758 REVIEW OF ECONOMIC STUDIES growth formula (7), and c(0)=0 by definition. How, though, can we get control on the enormous range of possible consumption processes and CGFs? One approach is to exploit Fact 1. CGFs are convex. Proof Since c(θ)=logm(θ), we have c (θ) = m(θ) m (θ) m (θ) 2 m(θ) 2 = EeθG EG 2 e θg ( EGe θg) 2 m(θ) 2. The numerator of this expression is positive by a version of the Cauchy Schwartz inequality that states that EX 2 EY 2 E( XY ) 2 for any random variables X and Y. In this case, we need to set X =e θg/2 and Y =Ge θg/2. (See Billingsley (1995), for further discussion of this and other properties of CGFs.) This fact can be used to derive sharp preference parameter bounds based on observables. Result 3. In the power utility case, we have r f c/w c/w ρ γ 1 rp+r f c/w. (11) In the Epstein Zin case, we have r f c/w c/w ρ 1/ψ 1 rp+r f c/w. (12) Moreover, these bounds are sharp: for any ε>0 there are distributions of consumption growth, and parameter choices for ρ, γ, and ψ that are consistent with the observables and satisfy c/w ρ 1/ψ 1 <r f c/w+ε; and other distributions of consumption growth and parameter choices that are consistent with the observables and satisfy c/w ρ 1/ψ 1 >rp+r f c/w ε. Proof The result is best understood geometrically. Look at Figure 5, which plots a generic CGF: it is convex and passes through the origin. We know from Result 2 that asset prices provide information about some key points on the CGF c( γ ), c(1 γ ), and c(1) and therefore about the gradients of the dashed lines marked X, Y, and Z. Now, because c(θ) is convex, the gradients of X, Y, and Z, in that order, are increasing: c(1 γ ) c( γ ) (1 γ ) ( γ ) These inequalities can be rearranged to give c(0) c(1 γ ) 0 (1 γ ) But from Equation (5) we have, in the Epstein Zin case, c(1) c(0). 1 0 c( γ ) γ c(1 γ ) c(1). (13) 1 γ c/w ρ 1/ψ 1 = c(1 γ ) 1 γ. (14)

15 IAN MARTIN CONSUMPTION-BASED ASSET PRICING 759 (a) Figure 5 Convexity of the CGF implies that line X has the smallest slope and line Z the largest (b) Figure 6 CGFs of two possible distributions of consumption growth that are consistent with the observables and illustrate why Putting (13) and (14) together, we have the bounds in Result 3 cannot be improved c( γ ) γ c/w ρ 1/ψ 1 c(1). The result (12) follows on rearranging the left-hand inequality using (4) and (5), and substituting out c(1) using the Gordon growth model; and (11) is a special case of (12). The detailed proof that the bounds are sharp is in the Appendix, but the main idea can be understood by comparing the generic CGF depicted in Figure 5 to the two CGFs shown in Figure 6, which indicate how the two extremes can be approximately attained. The Appendix demonstrates that there are distributions of consumption growth whose CGFs look like those in Figure 6 i.e. have the properties that (i) they are almost linear on the intervals [ γ,0] in the case represented by Figure 6a, or on [1 γ,1] in the case represented by Figure 6b; and (ii) they are consistent with the observables, which pins down c(1) and the gap c(1 γ ) c( γ ). The intuition is that as ψ approaches one, the consumption wealth ratio approaches ρ. Therefore, if the consumption wealth ratio is to be far from ρ, ψ must be far from one. Result 3 turns this qualitative statement into a quantitative one, without making assumptions about what is going on in the tails. Using the values rp=6%,r f =2%,c/w=6%, we have the restriction that

16 760 REVIEW OF ECONOMIC STUDIES (a) (b) Figure 7 Parameter restrictions for i.i.d. models with rp=6%, r f =2%, and c/w=6% 0.04 (0.06 ρ)/(1/ψ 1) The shaded areas in Figure 7 illustrate where the parameters must lie. If ψ>1, then ρ is constrained to lie between 0.02 and 0.08; if also ψ<2, then ρ must lie between 0.04 and If γ =1 in the power utility case, or if ψ =1 in the Epstein Zin case, then ρ is exactly identified by the consumption wealth ratio. To the extent that D t =C λ t is a reasonable approximation of leverage, we can say even more. For, we observe the consumption wealth ratio c/w, wealth risk premium rp w and the dividend yield on the market d/p and market risk premium rp m (that is, observe (4) (6), together with the expressions that result on substituting in λ=1). The following relationships hold: By convexity, we have c/w d/p = c(λ γ ) c(1 γ ) ϑ(ρ c/w)+c/w d/p = c(λ γ ) rp m +r f d/p = c(λ) c(λ γ ) c(1 γ ) λ 1 Substituting in, we have joint bounds on ρ, γ, and ψ: c/w d/p λ 1 ϑ(ρ c/w)+c/w d/p λ γ c(λ γ ) λ γ c(λ) λ. rp m +r f d/p. λ 3.1. Hansen Jagannathan and good-deal bounds Hansen and Jagannathan (1991) derived a bound that relates the standard deviation and mean of the stochastic discount factor, M, to the Sharpe ratio on an arbitrary asset, SR: SR σ (M) EM. (15) In the Epstein Zin-i.i.d. setting, the right-hand side of (15) becomes σ (M) EM = EM 2 (EM) 2 1 = e c( 2γ ) 2c( γ ) 1. (16)

17 IAN MARTIN CONSUMPTION-BASED ASSET PRICING 761 Combining (15) and (16), we obtain the Hansen Jagannathan bound in CGF notation: ( log 1+SR 2) c( 2γ ) 2c( γ ). (17) Cochrane and Saá-Requejo (2000) observe that inequality (15) suggests a natural way to restrict asset-pricing models. Suppose σ (M)/EM h; then (15) implies that the maximal Sharpe ratio is less than h. In CGF notation, the good-deal bound is written ( c( 2γ ) 2c( γ ) log 1+h 2). (18) Suppose, for example, that we wish to impose the restriction that Sharpe ratios above 1 are too good a deal to be available. Then the good-deal bound is c( 2γ ) 2c( γ ) log2. This expression can be evaluated under particular parametric assumptions about the consumption process. In the case in which consumption growth is lognormal, with volatility of log consumption equal to σ, it supplies an upper bound on risk aversion: γ log2/σ (which is about 42 if σ =0.02). However, this upper bound is rather weak, and in any case the postulated consumption process is inconsistent with observed features of asset markets such as the high equity premium and low riskless rate. Alternatively, one might model the consumption process as subject to disasters in the sense of Section 2.1. In this case, the good-deal bound implies tighter restrictions on γ, but these restrictions are sensitively dependent on the disaster parameters. In order to progress from (18) to a bound on γ and ρ that does not require parametrization of the consumption process, we want to relate c( 2γ ) 2c( γ ) to quantities that can be directly observed. For example, the Hansen Jagannathan bound (17) improves on a conclusion that follows from the convexity of the CGF, namely, that 0 c( 2γ ) 2c( γ ). (19) This trivial inequality follows by considering the value of the CGF at the three points c(0), c( γ ), and c( 2γ ). Convexity implies that the average slope of the CGF is more negative (or less positive) between 2γ and γ than it is between γ and 0: c( γ ) c( 2γ ) γ c(0) c( γ ). γ Equation (19) follows immediately, given that c(0)=0. Combining (18) and (19), we obtain the (underwhelming!) result that 0 log ( 1+h 2). However, we can sharpen (19) by comparing the slope of the CGF between 2γ and γ to the slope between γ and 1 γ (rather than between γ and 0): c( γ ) c( 2γ ) γ c(1 γ ) c( γ ). 1 This implies, by Result 1, that c( 2γ ) 2c( γ ) (γ 1)(c/w r f )+ϑ(c/w ρ), and hence Result 4. If the maximal Sharpe ratio is less than or equal to h, then we must have ( (γ 1)(c/w r f )+ϑ(c/w ρ) log 1+h 2). (20)

18 762 REVIEW OF ECONOMIC STUDIES (a) (b) Figure 8 Shaded areas indicate admissible parameter values for i.i.d. models with rp=6%, r f =2%, c/w=6%, and a maximal Sharpe ratio of 0.75 The important feature of this result is that by exploiting the observable consumption wealth ratio and riskless rate, we do not need to take a stand on what is going on in the tails. Figure 8 reproduces the bounds of Figure 7, adding in the good deal bound (20) with h=0.75, i.e. ruling out Sharpe ratios above Shaded areas indicate admissible parameter values. In Figure 8b, admissible values lie below the line marked good deal bound when ψ<1 and above it when ψ>1. There are also admissible values of ρ and ψ not visible in the figure, with ρ large and ψ close to zero. The figure assumes γ =4, but the admissible regions are unaltered for any value of γ between 0 and 8.5. The line marked good deal bound steepens as γ increases, while continuing to pass through the point (0.06,1); once γ rises above 8.5, the good deal bound starts to impose tighter constraints on ψ and ρ. I repeat this exercise in the Supplementary Appendix for a maximum Sharpe ratio of 1.25 rather than With power utility, this increase in h shifts the good deal bound upwards, meaning that higher values of γ are permissible for fixed ρ. With Epstein Zin preferences, the good deal bound flattens out, which has no effect on the visible admissible region if γ is held constant at EXTENSIONS 4.1. Log consumption not proportional to log dividends Thus far we have operated under the assumption that log dividends are proportional to log consumption. This section considers more general scenarios in which consumption and dividends may differ, with a motivating application to a heterogeneous-agent economy. Again, the goal will be to make statements that are independent of particular assumptions about tail behaviour. To introduce some notation, suppose that a power utility agent with consumption C t is pricing an asset paying dividends D t, so that the asset s price satisfies ( ) γ P 0 =D 0 E e ρt Ct Dt dt. t=0 C 0 D 0 Defining the bivariate CGF c(θ 1,θ 2 ) logee θ 1log(C t+1 /C t )+θ 2 log(d t+1 /D t ), it is almost immediate, following the same logic as before, that D 0 /P 0 =ρ c( γ,1). The next result, whose proof is straightforward, collects this fact together with the riskless rate and risk premium.

19 IAN MARTIN CONSUMPTION-BASED ASSET PRICING 763 Result 5. If consumption growth G t logc t /C 0 and dividend growth H t logd t /D 0 are distinct, potentially correlated, Lévy processes, then D 0 /P 0 = ρ c( γ,1) R f = ρ c( γ,0) RP = c(0,1)+c( γ,0) c( γ,1). Constantinides and Duffie (1996) have shown that accounting for heterogeneity may contribute to an understanding of the equity premium puzzle. On the other hand, Grossman and Shiller (1982) have shown that if agents consumption processes follow diffusions, risk premia are unaffected by heterogeneity. The tension between these two results will be resolved by showing that heterogeneity matters to the extent that it is present at times of aggregate jumps. The presence of jumps lends a discrete-time flavor to the model, and as a result it lies closer on the spectrum to Constantinides Duffie than to Grossman Shiller. I assume that agents suffer idiosyncratic shocks to consumption, perhaps because agents have labour income risk that is uninsurable for moral hazard reasons. 6 All agents have power utility. Agent i s log consumption process is given by log C N t i,t = μt +σ B t + Y j C i,0 j=1 }{{} common to all agents +σ 1 B it 1 2 σ 1 2 t }{{} type (i) N i,t + N t X i,j + Y i,j j=1 j=1 }{{}}{{} type (ii) type (iii) Here B t is a Brownian motion, and Y j are the i.i.d. sizes of jumps, which occur at times dictated by the Poisson process N t, with arrival rate ω; these shocks are common to all agents. I assume that the jumps are bad or at least, not good news on average, so Ee Y j 1. There are also three types of idiosyncratic shocks: (i) an idiosyncratic Brownian motion component, B i,t ; (ii) jumps whose size X i,k and timing (determined by the Poisson process N i,t ) are idiosyncratic; and (iii) jumps of idiosyncratic size Y i,k, whose timing coincides with aggregate disasters, to capture the fact that disasters do not have the same impact on all agents. I assume that X i,j and Y k,l are i.i.d. across i,j,k and l, and N i,t are Poisson processes, independent across i, with arrival rate ω 2. Finally, σ 1 and ω 2 are constant across all agents i. Aggregate quantities are computed by summing over agents i; I assume that a law of large numbers holds so that this process is equivalent to taking an expectation over i. With this assumption, together with the normalization that for all i and k, Ee X i,k =Ee Y i,k =1, (21) implies that aggregate consumption evolves according to log C t C 0 =μt +σ B t + N t j=1 Y j. The upshot is that all agents attach the same value to the equity claim to aggregate consumption, so as in Constantinides and Duffie (1996) there is a no-trade equilibrium in which agent i consumes C i,t at time t. The Euler equation holds for each agent i, so the price of equity, P, must satisfy ( ) γ P =E e ρt Ci,t C t dt. (22) 0 C i,0 Result 5 now applies with G t =logc it /C i0 and H t =logc t /C 0. Defining m D (θ) Ee θy j, m 2 (θ) Ee θx i,k, and m 3 (θ) Ee θy i,j, the CGF of (G 1,H 1 )isc(θ 1,θ 2 )=μ(θ 1 +θ 2 )+ 1 2 σ 2 (θ 1 +θ 2 ) 2 + (21) 6. Storesletten, Telmer, and Yaron (2004) show that idiosyncratic shocks are highly persistent, and large, with a standard deviation of about 0.25.